On the initial value problem of the semilinear beam equation with weak damping I: Smoothing effect

MSC: 35G10 35B65 35G25 35E05 Keywords: Damped beam equation Cauchy problem Fourth order wave equation with damping Asymptotic profile PerturbationWe study the initial value problem for some semilinear damped beam equation. In Takeda and Yoshikawa (submitted for publication) [9] unique global existence of a decaying solution for the problem and the smoothing effect of the solution was shown. In this article we shall give the asymptotic profiles of the solution. As a result, we observe that the decay estimates are optimal. Moreover, considering the higher order expansion of the solution, we observe more detailed information such as the contribution of the nonlinear term to the solution as t → ∞.

show abstract

“…The paper is a sequel to Part I [9]. We refer to the introduction in Part I for the motivations for studying (1.1)-(1.2).…”

Section: Introductionmentioning

confidence: 99%

“…In [9] unique global existence of a solution to (1.1)-(1.2) and the smoothing effect of the solution were established.…”

Section: Introductionmentioning

confidence: 99%

On the initial value problem of the semilinear beam equation with weak damping II: Asymptotic profiles

Takeda

Yoshikawa

2012

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…The inequalities (2.6) and (2.7) can be found in Propositions 3.1 and 3.2 in [11] as a special case (q = r = 2). Therefore we only show (2.8) and (2.9).…”

Section: Preliminariesmentioning

confidence: 81%

“…Although we could derive L p -L q estimate by using the Carlson-Beurling inequality as in [7] and [11], we only state L 2 -L 2 case for simplicity because we use only this case in this article. Proposition 2.3.…”

Section: Preliminariesmentioning

confidence: 99%

Decay estimates for the Cauchy problem for the damped extensible beam equation

Racke

Yoshikawa

2015

Applicable Analysis

Self Cite

View full text Add to dashboard Cite

The extensible beam equation proposed by Woinovsky-Krieger [13] is a fourth order dispersive equation with nonlocal nonlinear terms. In this paper we study the Cauchy problem of the extended model by Ball who proposed the following model with external and structural damping terms:For η > 0 this represents a Kelvin-Voigt damping. We show the unique global existence of solution for this problem and give a precise description of the decay of solutions in time.

show abstract

“…The proof of such results can be done in a similar way as in Theorems 3.1 and 3.2. Recently, asymptotic profiles and some other properties of solutions for this type of equations are studied in [19] at least in the one dimensional case. Theorem 3.3.…”

Section: Proof Multiplying Equation (33) By P (ξ) −1 We Obtainmentioning

confidence: 99%

Asymptotic behavior for abstract evolution differential equations of second order

Luz

Ikehata

Charão

2015

Journal of Differential Equations

View full text Add to dashboard Cite

evolution differential equations of second order in time are studied in order to get (almost) optimal decay estimates to the corresponding energy functional of the equations. The framework is supported by a special energy method in the associated Fourier space. The constructed abstract theory can be applied to several concrete evolutionary partial differential equations as is illustrated in the last section of the paper.

show abstract

On the initial value problem of the semilinear beam equation with weak damping I: Smoothing effect

Cited by 12 publications

References 20 publications

On the initial value problem of the semilinear beam equation with weak damping II: Asymptotic profiles

On the initial value problem of the semilinear beam equation with weak damping II: Asymptotic profiles

Decay estimates for the Cauchy problem for the damped extensible beam equation

Asymptotic behavior for abstract evolution differential equations of second order

Contact Info

Product

Resources

About